Studies Of Optimization Problem About Fuzzy Relation

The feasible domain of an optimization model subject to a system of fuzzy relations equations is the solution set of fuzzy relation equations. When the solution set of fuzzy relation equations is non-empty, the feasible domain is, in general, a non-convex set, which can be completely determined by a maximum solution and a finite number of minimum solutions. Because the feasible domain is non-convex, traditional linear programming methods, such as the simplex, become useless. In 1991, a new kind of optimization theory, latti-cized linear programming subject to fuzzy relation inequalities, was proposed by Wang Pei-zhuang, then Wang Pei-zhuang solved all minimum solutions by conservative path methods to find the optimal solution of the optimization model with minimum operation for object. Then, Fang Shu-cherng studied programming problem with a linear objective function subject to a system of fuzzy relation equations about, max — min composition, then convert the problem to an equivalent involving 0-1 integer programming with a branch-and-bound solution technique to find an optimal solution. In his paper, we present a new method, min-max methods, to solving latticized linear programming based on fuzzy relation equation with max-product composition. By this methods, we without changing problem to an equivalent, involving 0-1 integer programming or obtaining all minimum solutions to fuzzy relation equations can solve all the minimum solutions for finding an optimal solution of problem. At the same time, the optimal solution of the latticized linear programming with max-product composition is several minimum solutions or the subset of the feasible domain. In this paper, we find the all optimal solution sets of latticized linear programming with max-product by min-max methods. And we build similar min-max methods for an optimal solution of latticized linear programming with ∨-∧ composition. At last, we study latticized linear programming with index in the objective function and max-product composition, then build max methods to find an optimal solution of minimum objective function and similar min-max methods to find an optimal solution of maximum objective function.